Tuesday, May 15, 2018

Unizor - Physics4Teens - Mechanics - Dynamics - Superposition of Forces ...

Notes to a video lecture on http://www.unizor.com

Superposition of Forces -

Problems 1

Problem A

A boat stands still in the middle of a river. It is tied to a river bank by a rope of the length L.

The river flow pushes it with the force F downstream, and the wind pushes it with the force 0.75·F perpendicularly to the river flow away from the bank.

(a) What is the force of tension P of the rope?

(b) What is the distance d from the river bank to the boat?


(a) P = 1.25·F

(b) d = 0.6·L

Problem B

An object of mass m and, therefore, of weight at the ground W=m·g, where g
is a free fall acceleration, moves with constant linear speed on a
bridge, that has a shape of an arc with upward convexity of a radius R.

(a) Determine the speed v of an object as a function of pressure P it produces on the bridge at the top of the arc.

(b) At what speed v0 the object will become "weightless" at the top of the bridge's arc?


(a) v = √R·(g − P/m)

(b) v0 = √R·g,

because "weightlessness" is a state when an object does not press on the support and, therefore, P=0

Problem C

A point object of mass m and, therefore, of weight W=m·g, where g is a free fall acceleration, is hanging on the end of a weightless thread of length R,
fixed at the other end at some point, and can freely move like a
pendulum within a vertical plane that we can take as the XZ-plane of
coordinates with origin at the point where a thread is fixed.

An object's initial position is with its thread making an angle φ from a vertical.

(a) What is the tension of a thread P in the initial position?

(b) What is the vector of force F acting on an object in its initial position in the direction of its motion along a circular trajectory?


(a) Tension P = m·g·cos(φ)

(b) Force along the trajectory F = m·g·sin(φ)

Problem D

A projectile is launched from the wall of the castle horizontally towards the army that laid a siege on this castle.

The height of the castle wall is H, the horizontal speed of a projectile is v.

The free falling acceleration caused by gravity is g.

Ignore the air resistance.

(a) What is the time tLand the projectile will be in the air until landing?

(b) What is the distance d of a point the projectile lands from the bottom of the castle wall?

(c) What is the magnitude of a speed of the projectile vLand at the moment of landing?


(a) tLand = 2H/g

(b) d = v·√2H/g

(c) vLand = √v² + 2Hg

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